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Lecture_10__Finite_Square_Well.pdf
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Page 20
20
Normalize wavefunction …
/
)
sin(
)
(
)
(
)
,
(
iEt
e
L
x
n
B
t
x
t
x
1
)
/
(
sin

)
,
(

2
0
2
0
*
2
dx
L
x
n
B
dx
dx
t
x
L
L
L
B
2
Probability of finding electron between 
∞
and
∞
must be 1.
/
)
sin(
2
)
(
)
(
)
,
(
iEt
e
L
x
n
L
t
x
t
x
Solving completely
Page 21
21
Quantized: k=n
/L
Quantized:
1
2
2
2
2
2
2
E
n
mL
n
E
show sim vary L, n
energies are
“eigen values”
wave functions
“eigenfunctions”
/
)
sin(
2
)
,
(
iEt
e
L
x
n
L
t
x
Page 22
22
/
)
sin(
2
)
,
(
iEt
e
L
x
n
L
t
x
Quantized: k=n
/L
Quantized:
1
2
2
2
2
2
2
E
n
mL
n
E
n=2
What you expect classically:
Electron can have any energy
Electron is localized
Electron is delocalized
… spread out between 0 and L
What you get quantum
mechanically:
Electron can only have specific
energies. (quantized)
Page 23
23
CQ.
Can an electron be in the
superposition of
n=1 and n=3 eigenstates?
a. yes,
b. no
CQ. if we measured the position of this electron and
found it at x =2+.01 nm, what its energy would be?
think, then will give choices.
a. one particular discrete energy but cannot know which.
b. sum of many different energies.
c. would be impossible to do the measurement
d. the energy for which wave function peaked at that spot
e. some other answer
E
x
Page 24
24
When does it fail the clever physicist model based on a
wavefunction that never gets outside potential well?
(look at sim)
Page 25
25
Good Approximation:
Electrons never got out of wire
(x<0 or x>L) =0.
(OK when Energy << work function)
Exact Potential Energy curve:
V. small chance electrons get out of
wire
(x<0 or x>L)~0, but not exactly 0!
0
Energy
x
0
L
V(x)
What happens if electron Energy bigger?
What if two wires very close to each other?
Then if
leaks out a little or not, is very important.
How much coupling to other wire?
E_total
Page 26
When does fail the clever physicist model based on a
wavefunction that never gets outside potential well?
26
If energy of electron not too far below top of well, starts to
extend out a bit.
(classically forbidden region)
Some special situations are VERY important.
a. two wires close together
b. nuclear decay,
c. tunneling microscope see single atoms on surfaces
d. most modern electronics
e. electrons coming out of hot metal
f
....
Page 27
27
Detect electron position see it outside wire?
Page 28
28
sketching how to solve for wave function for finite
potential wells (boundry conditions are challenging).
V. small chance electrons get out of wire
(x<0 or x>L)~0, but not exactly 0!
wire
0
Energy
x
0
L
V(x)
Finite
Square
Well
Workfunction
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